Timeline for Uniform continuity of heat semigroup
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Apr 26, 2021 at 10:36 | vote | accept | Sascha | ||
| Feb 11, 2021 at 1:11 | comment | added | Migalobe | Sorry, I'm a bit confused. If $T(t)$ on $L^2(\mathbb{R}^n)$ is strongly continuous, this wouldn't contradict the non uniqueness of solutions? | |
| May 30, 2018 at 11:46 | answer | added | Bazin | timeline score: 0 | |
| May 29, 2018 at 16:09 | comment | added | Jochen Glueck | Does Theorem 2 in my answer below address your intention of getting a general understanding of the situation, or are you more interested in semigroups whose generator does not have compact resolvent (as, e.g., the heat semigroup on the entire space $\mathbb{R}^n$ which is your motivating example). | |
| May 27, 2018 at 18:51 | answer | added | Jochen Glueck | timeline score: 2 | |
| May 23, 2018 at 17:36 | comment | added | Jochen Glueck | Well, if we would like to be bold, we could conjecture that every subspace $X$ with the desired property is a direct sum of a subspace of $D(A)$ (which is, in addition, closed in $X$) and a finite-dimensional space. Unfortunately though, experience thaught me better not to be bold... | |
| May 23, 2018 at 17:14 | comment | added | Sascha | @JochenGlueck thank you, that's right. However, I introduced these two examples only to illustrate what is possible and that it is not clear to me what will happen for arbitrary spaces. | |
| May 23, 2018 at 16:08 | comment | added | Jochen Glueck | If $X\subset D(A)$, then the semigroup restricted to $X$ is uniformly continuous; this follows by applying the uniform boundedness principle to the restriction of the differential quotient $(T(t) - I)/t$ to $X$. This observation includes your spectral measure construction as a special case, but it does not include the case where $X$ is finite dimensional. | |
| May 23, 2018 at 14:35 | history | edited | Sascha | CC BY-SA 4.0 |
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| May 23, 2018 at 13:27 | history | edited | Sascha | CC BY-SA 4.0 |
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| May 23, 2018 at 11:17 | history | edited | Sascha | CC BY-SA 4.0 |
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| May 23, 2018 at 11:02 | history | asked | Sascha | CC BY-SA 4.0 |